Auction Theory II

Recap First-Price Revenue Equivalence Optimal Auctions

Auction Theory II

Lecture 19

Auction Theory II Lecture 19, Slide 1 Recap First-Price Revenue Equivalence Optimal Auctions Lecture Overview

1 Recap

2 First-Price Auctions

3 Revenue Equivalence

4 Optimal Auctions

Auction Theory II Lecture 19, Slide 2 Recap First-Price Revenue Equivalence Optimal Auctions Motivation

Auctions are any mechanisms for allocating resources among self-interested agents resource allocation is a fundamental problem in CS increasing importance of studying distributed systems with heterogeneous agents currency needn’t be real money, just something scarce

Auction Theory II Lecture 19, Slide 3 Recap First-Price Revenue Equivalence Optimal Auctions Intuitive comparison of 5 auctions

Intuitive Comparison of 5 auctions

English Dutch Japanese 1st-Price 2nd-Price

Duration #bidders, starting #bidders, fixed fixed increment price, clock increment speed Info 2nd-highest winner’s all val’s but none none Revealed val; bounds bid winner’s on others Jump bids yes n/a no n/a n/a

Price yes no yes no no Discovery Fill in “regret” after Regret no yes no yes no the fun game

Auction• Theory How II should agents bid in these auctions? Lecture 19, Slide 4 Recap First-Price Revenue Equivalence Optimal Auctions Second-Price proof

Theorem Truth-telling is a dominant strategy in a second-price auction.

Proof. Assume that the other bidders bid in some arbitrary way. We must show that i’s best response is always to bid truthfully. We’ll break the proof into two cases: 1 Bidding honestly, i would win the auction 2 Bidding honestly, i would lose the auction

Auction Theory II Lecture 19, Slide 5 Theorem Under the independent private values model (IPV), it is a dominant strategy for bidders to bid up to (and not beyond) their valuations in both Japanese and English auctions.

Recap First-Price Revenue Equivalence Optimal Auctions English and Japanese auctions

A much more complicated strategy space extensive form game bidders are able to condition their bids on information revealed by others in the case of English auctions, the ability to place jump bids intuitively, though, the revealed information doesn’t make any diﬀerence in the IPV setting.

Auction Theory II Lecture 19, Slide 6 Recap First-Price Revenue Equivalence Optimal Auctions English and Japanese auctions

Theorem Under the independent private values model (IPV), it is a dominant strategy for bidders to bid up to (and not beyond) their valuations in both Japanese and English auctions.

Auction Theory II Lecture 19, Slide 6 Recap First-Price Revenue Equivalence Optimal Auctions Lecture Overview

1 Recap

2 First-Price Auctions

3 Revenue Equivalence

4 Optimal Auctions

Auction Theory II Lecture 19, Slide 7 Recap First-Price Revenue Equivalence Optimal Auctions First-Price and Dutch

Theorem First-Price and Dutch auctions are strategically equivalent.

In both ﬁrst-price and Dutch, a bidder must decide on the amount he’s willing to pay, conditional on having placed the highest bid. despite the fact that Dutch auctions are extensive-form games, the only thing a winning bidder knows about the others is that all of them have decided on lower bids e.g., he does not know what these bids are this is exactly the thing that a bidder in a ﬁrst-price auction assumes when placing his bid anyway. Note that this is a stronger result than the connection between second-price and English.

Auction Theory II Lecture 19, Slide 8 They should clearly bid less than their valuations. There’s a tradeoﬀ between: probability of winning amount paid upon winning Bidders don’t have a dominant strategy any more.

Recap First-Price Revenue Equivalence Optimal Auctions Discussion

Auction Theory II Lecture 19, Slide 9 Recap First-Price Revenue Equivalence Optimal Auctions Discussion

So, why are both auction types held in practice? First-price auctions can be held asynchronously Dutch auctions are fast, and require minimal communication: only one bit needs to be transmitted from the bidders to the auctioneer. How should bidders bid in these auctions? They should clearly bid less than their valuations. There’s a tradeoﬀ between: probability of winning amount paid upon winning Bidders don’t have a dominant strategy any more.

Auction Theory II Lecture 19, Slide 9 Recap First-Price Revenue Equivalence Optimal Auctions Analysis

Theorem

In a ﬁrst-price auction with two risk-neutral bidders whose valuations are drawn 1 1 independently and uniformly at random from [0, 1], ( 2 v1, 2 v2) is a Bayes-Nash equilibrium strategy proﬁle.

Proof. 1 Assume that bidder 2 bids 2 v2, and bidder 1 bids s1. From the fact that v2 was drawn from a uniform distribution, all values of v2 between 0 and 1 are equally likely. Bidder 1’s expected utility is

Z 1 E[u1] = u1dv2. (1) 0 Note that the integral in Equation (1) can be broken up into two smaller integrals that diﬀer on whether or not player 1 wins the auction.

Z 2s1 Z 1 E[u1] = u1dv2 + u1dv2 0 2s1

Auction Theory II Lecture 19, Slide 10 Recap First-Price Revenue Equivalence Optimal Auctions Analysis

Theorem

In a ﬁrst-price auction with two risk-neutral bidders whose valuations are drawn 1 1 independently and uniformly at random from [0, 1], ( 2 v1, 2 v2) is a Bayes-Nash equilibrium strategy proﬁle.

Proof (continued).

1 We can now substitute in values for u1. In the ﬁrst case, because 2 bids 2 v2, 1 wins when v2 < 2s1, and gains utility v1 − s1. In the second case 1 loses and gains utility 0. Observe that we can ignore the case where the agents have the same valuation, because this occurs with probability zero.

Z 2s1 Z 1 E[u1] = (v1 − s1)dv2 + (0)dv2 0 2s1 2s1

= (v1 − s1)v2 0 2 = 2v1s1 − 2s1 (2)

Auction Theory II Lecture 19, Slide 10 Recap First-Price Revenue Equivalence Optimal Auctions Analysis

Theorem

In a ﬁrst-price auction with two risk-neutral bidders whose valuations are drawn 1 1 independently and uniformly at random from [0, 1], ( 2 v1, 2 v2) is a Bayes-Nash equilibrium strategy proﬁle.

Proof (continued). We can ﬁnd bidder 1’s best response to bidder 2’s strategy by taking the derivative of Equation (2) and setting it equal to zero:

∂ 2 (2v1s1 − 2s1) = 0 ∂s1

2v1 − 4s1 = 0 1 s = v 1 2 1 Thus when player 2 is bidding half her valuation, player 1’s best strategy is to bid half his valuation. The calculation of the optimal bid for player 2 is analogous, given the symmetry of the game and the equilibrium.

Auction Theory II Lecture 19, Slide 10 Recap First-Price Revenue Equivalence Optimal Auctions More than two bidders

Very narrow result: two bidders, uniform valuations. Still, ﬁrst-price auctions are not incentive compatible hence, unsurprisingly, not equivalent to second-price auctions

Theorem In a ﬁrst-price sealed bid auction with n risk-neutral agents whose valuations are independently drawn from a uniform distribution on the same bounded interval of the real numbers, the unique symmetric equilibrium is given by the strategy proﬁle n−1 n−1 ( n v1,..., n vn).

proven using a similar argument, but more involved calculus a broader problem: that proof only showed how to verify an equilibrium strategy. How do we identify one in the ﬁrst place?

Auction Theory II Lecture 19, Slide 11 Recap First-Price Revenue Equivalence Optimal Auctions Lecture Overview

1 Recap

2 First-Price Auctions

3 Revenue Equivalence

4 Optimal Auctions

Auction Theory II Lecture 19, Slide 12 Recap First-Price Revenue Equivalence Optimal Auctions Revenue Equivalence

Which auction should an auctioneer choose? To some extent, it doesn’t matter...

Theorem (Revenue Equivalence Theorem)

Assume that each of n risk-neutral agents has an independent private valuation for a single good at auction, drawn from a common cumulative distribution F (v) that is strictly increasing and atomless on [v, v¯]. Then any auction mechanism in which the good will be allocated to the agent with the highest valuation; and any agent with valuation v has an expected utility of zero; yields the same expected revenue, and hence results in any bidder with valuation v making the same expected payment.

Auction Theory II Lecture 19, Slide 13 Recap First-Price Revenue Equivalence Optimal Auctions Revenue Equivalence Proof

Proof.

Consider any mechanism (direct or indirect) for allocating the good. Let ui(vi) be i’s expected utility given true valuation vi, assuming that all agents including i follow their equilibrium strategies. Let Pi(vi) be i’s probability of being awarded the good given (a) that his true type is vi; (b) that he follows the equilibrium strategy for an agent with type vi; and (c) that all other agents follow their equilibrium strategies.

ui(vi) = viPi(vi) − E[payment by type vi of player i] (1)

From the deﬁnition of equilibrium, for any other valuation vˆi that i could have,

ui(vi) ≥ ui(ˆvi) + (vi − vˆi)Pi(ˆvi). (2)

To understand Equation (2), observe that if i followed the equilibrium strategy for a player with valuation vî rather than for a player with his (true) valuation vi, i would make all the same payments and would win the good with the same probability as an agent with valuation vî. However, whenever he wins the good, i values it (vi − vî) more than an agent of type vî does. The inequality must hold because in equilibrium this deviation must be unprofitable. Auction Theory II Lecture 19, Slide 14 Recap First-Price Revenue Equivalence Optimal Auctions Revenue Equivalence Proof

Proof (continued).

Consider vˆi = vi + dvi, by substituting this expression into Equation (2):

ui(vi) ≥ ui(vi + dvi) + dviPi(vi + dvi). (3)

Likewise, considering the possibility that i’s true type could be vi + dvi,

ui(vi + dvi) ≥ ui(vi) + dviPi(vi). (4)

Combining Equations (4) and (5), we have

ui(vi + dvi) − ui(vi) Pi(vi + dvi) ≥ ≥ Pi(vi). (5) dvi

dui Taking the limit as dvi → 0 gives = Pi(vi). Integrating up, dvi

Z vi ui(vi) = ui(v) + Pi(x)dx. (6) x=v

Auction Theory II Lecture 19, Slide 14 Recap First-Price Revenue Equivalence Optimal Auctions Revenue Equivalence Proof

Proof (continued). Now consider any two eﬃcient auction mechanisms in which the expected payment of an agent with valuation v is zero. A bidder with valuation v will never win (since the distribution is atomless), so his expected utility ui(v) = 0. Because both mechanisms are eﬃcient, every agent i always has the same Pi(vi) (his probability of winning given his type vi) under the two mechanisms. Since the right-hand side of Equation (6) involves only Pi(vi) and ui(v), each agent i must therefore have the same expected utility ui in both mechanisms. From Equation (1), this means that a player of any given type vi must make the same expected payment in both mechanisms. Thus, i’s ex ante expected payment is also the same in both mechanisms. Since this is true for all i, the auctioneer’s expected revenue is also the same in both mechanisms.

Auction Theory II Lecture 19, Slide 14 Thus in a second-price auction, the seller’s expected revenue is n − 1 v . n + 1 max

First and second-price auctions satisfy the requirements of the revenue equivalence theorem every symmetric game has a symmetric equilibrium in a symmetric equilibrium of this auction game, higher bid ⇔ higher valuation

Recap First-Price Revenue Equivalence Optimal Auctions First and Second-Price Auctions

The kth order statistic of a distribution: the expected value of the kth-largest of n draws. th For n IID draws from [0, vmax], the k order statistic is n + 1 − k v . n + 1 max

Auction Theory II Lecture 19, Slide 15 First and second-price auctions satisfy the requirements of the revenue equivalence theorem every symmetric game has a symmetric equilibrium in a symmetric equilibrium of this auction game, higher bid ⇔ higher valuation

Recap First-Price Revenue Equivalence Optimal Auctions First and Second-Price Auctions

The kth order statistic of a distribution: the expected value of the kth-largest of n draws. th For n IID draws from [0, vmax], the k order statistic is n + 1 − k v . n + 1 max

Thus in a second-price auction, the seller’s expected revenue is n − 1 v . n + 1 max

Auction Theory II Lecture 19, Slide 15 Recap First-Price Revenue Equivalence Optimal Auctions First and Second-Price Auctions

The kth order statistic of a distribution: the expected value of the kth-largest of n draws. th For n IID draws from [0, vmax], the k order statistic is n + 1 − k v . n + 1 max

Thus in a second-price auction, the seller’s expected revenue is n − 1 v . n + 1 max

Auction Theory II Lecture 19, Slide 15 Recap First-Price Revenue Equivalence Optimal Auctions Applying Revenue Equivalence

Thus, a bidder in a FPA must bid his expected payment conditional on being the winner of a second-price auction this conditioning will be correct if he does win the FPA; otherwise, his bid doesn’t matter anyway if vi is the high value, there are then n − 1 other values drawn from the uniform distribution on [0, vi] thus, the expected value of the second-highest bid is the ﬁrst-order statistic of n − 1 draws from [0, vi]:

n + 1 − k (n − 1) + 1 − (1) n − 1 v = (v ) = v n + 1 max (n − 1) + 1 i n i This provides a basis for our earlier claim about n-bidder ﬁrst-price auctions. However, we’d still have to check that this is an equilibrium The revenue equivalence theorem doesn’t say that every revenue-equivalent strategy proﬁle is an equilibrium!

Auction Theory II Lecture 19, Slide 16 Recap First-Price Revenue Equivalence Optimal Auctions Lecture Overview

1 Recap

2 First-Price Auctions

3 Revenue Equivalence

4 Optimal Auctions

Auction Theory II Lecture 19, Slide 17 Recap First-Price Revenue Equivalence Optimal Auctions Optimal Auctions

So far we have only considered eﬃcient auctions. What about maximizing the seller’s revenue? she may be willing to risk failing to sell the good even when there is an interested buyer she may be willing sometimes to sell to a buyer who didn’t make the highest bid Mechanisms which are designed to maximize the seller’s expected revenue are known as optimal auctions.

Auction Theory II Lecture 19, Slide 18 Recap First-Price Revenue Equivalence Optimal Auctions Optimal auctions setting

independent private valuations risk-neutral bidders each bidder i’s valuation drawn from some strictly increasing cumulative density function Fi(v) (PDF fi(v)) we allow Fi 6= Fj: asymmetric auctions

the seller knows each Fi

Auction Theory II Lecture 19, Slide 19 Theorem The optimal (single-good) auction is a sealed-bid auction in which every agent is asked to declare his valuation. The good is sold to ∗ the agent i = arg maxi ψi(ˆvi), as long as vi > ri . If the good is sold, the winning agent i is charged the smallest valuation that he could have declared while still remaining the winner: ∗ ∗ ∗ inf{vi : ψi(vi ) ≥ 0 and ∀j 6= i, ψi(vi ) ≥ ψj(ˆvj)}.

Recap First-Price Revenue Equivalence Optimal Auctions Designing optimal auctions

Deﬁnition (virtual valuation)

1−Fi(vi) Bidder i’s virtual valuation is ψi(vi) = vi − . fi(vi)

Definition (bidder-specific reserve price) ∗ Bidder i’s bidder-specific reserve price ri is the value for which ∗ ψi(ri ) = 0.

Auction Theory II Lecture 19, Slide 20 Recap First-Price Revenue Equivalence Optimal Auctions Designing optimal auctions

Deﬁnition (virtual valuation)

1−Fi(vi) Bidder i’s virtual valuation is ψi(vi) = vi − . fi(vi)

Definition (bidder-specific reserve price) ∗ Bidder i’s bidder-specific reserve price ri is the value for which ∗ ψi(ri ) = 0.

Theorem The optimal (single-good) auction is a sealed-bid auction in which every agent is asked to declare his valuation. The good is sold to ∗ the agent i = arg maxi ψi(ˆvi), as long as vi > ri . If the good is sold, the winning agent i is charged the smallest valuation that he could have declared while still remaining the winner: ∗ ∗ ∗ inf{vi : ψi(vi ) ≥ 0 and ∀j 6= i, ψi(vi ) ≥ ψj(ˆvj)}.

Auction Theory II Lecture 19, Slide 20 No, it’s not eﬃcient. How should bidders bid? it’s a second-price auction with a reserve price, held in virtual valuation space. neither the reserve prices nor the virtual valuation transformation depends on the agent’s declaration thus the proof that a second-price auction is dominant-strategy truthful applies here as well.